The ULN2003 looks a lot better.Now the matrix is not wired like the data sheet is it.The data sheet shows pins 9, 14, 12 & 8, 1, 7, 2 all being anodes. These anodes should be connected to the arduino outputs. The data sheet shows pins 13, 3, 4 & 11, 10, 6 as all being cathodes these should be connected to your ULN2003.

If you look on the bottom of the part you may see a 1 next to one of the corner pins.Plug it in and drive the output pin for C1 high (display pin 13) and for R1 low (display pin 9) - should see the corner LED turn on. If not, rotate the display 180 degrees around and try again.